Is there a way to grab this info via the API to display in my application?
Not easily, no. These are "ever changing" values and anything that TurboActivate would have would be immediately outdated.
You could, however, make a call to a script on your website from your app. The call would pass the customer's product key. And the script would use the web API (including http://wyday.com/limelm/help/api/limelm.pkey.getDetails/) to get the current details of a product key.
Does that make sense?